**Surface area and volume of a cube examples :**

Here we are going to see some examples problems on finding surface area and volume of a cube.

Let the side of a cube be a units. Then :

(i) The Total Surface Area (T.S.A) = 6a^{2} square units.

(ii) The Lateral Surface Area (L.S.A) = 4a^{2} square units.

(ii) Volume of cube = a^{3} cubic units.

**Example 1 :**

The volume of the cube is 125 dm^{3}. Find its side.

**Solution :**

Volume of cube = 125 dm^{3}

a^{3} = 125

a^{3} = 5^{3}

a = 5 dm

Hence side length of cube is 5 dm.

**Example 2 :**

A container is in the shape of a cube of side 20 cm. How much sugar can it hold?

**Solution :**

In order to find the quantity of sugar that the container can hold, we have to find the volume of container.

Side length of cubical container = 20 cm

Volume of container = a^{3}

= (20)^{3}

= 8000 cm^{3}

**Example 3 :**

A cubical tank can hold 64,000 litres of water. Find the length of the side of the tank.

**Solution :**

Let a be the side of the cubical tank. Volume of the tank is 27,000 litres. So,

V = a^{3} = (27000/1000)

a^{3} = 27

a = ∛27

a = 3 m

**Example 4 :**

Three metallic cubes of side 3 cm, 4 cm and 5 cm respectively are melted and are recast into a single cube. Find the total surface area of the new cube.

**Solution :**

Side length of 1^{st} cube = 3 cm

Side length of 2^{nd} cube = 4 cm

Side length of 3^{rd} cube = 5 cm

Volume of 1^{st}, 2^{nd} and 3^{rd} cube = 3^{3} + 4^{3} + 5^{3}

= 27 + 64 + 125

= 216

a^{3} = 216

a^{3} = 6^{3}

a = 6 cm

Total surface area of new cube = 6a^{2}

= 6(6)^{2}

= 6(36)

= 216 cm^{2}

Hence total surface area of new cube 216 cm^{2}

**Example 5 :**

Find the L.S.A, T.S.A and volume of a cube of side 5 cm.

**Solution :**

Lateral surface area (L.S.A) = 4a^{2}

= 4(52 ) = 100 sq. cm

Total surface area (T.S.A) = 6a^{2}

= 6 (5^{2} )

= 150 sq. cm

Volume of cube = a^{3}

= 5^{3}

= 125 cm^{3}

**Example 6 :**

Find the length of the side of a cube whose total surface area is 216 square cm.

**Solution :**

Let a be the side of the cube.

Given that T.S.A = 216 sq. cm

6a^{2} = 216

a^{2} = 216/6

a^{2} = 36

a = √36

a = 6 cm

**Example 7 :**

A cube has a total surface area of 384 sq. cm. Find its volume.

**Solution :**

Let a be the side of the cube. Given that T.S.A = 384 sq. cm

6a^{2} = 384

a^{2} = 384/6

a^{2} = 64

a = √64 = 8 cm

Hence, volume = a^{3} = 8^{3} = 512 cm^{3}

**Example 8 :**

If the lateral surface area of a cube is 900 cm^{2}, find the length of its side.

**Solution :**

Lateral surface area of cube = 4a^{2}

4a^{2 }= 900

a^{2 }= 900/4

a^{2 }= 225

a = √225

a = 15 cm

Hence the side of cube is 15 cm.

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